# Properties

 Label 17.1.861...144.1 Degree $17$ Signature $[1, 8]$ Discriminant $8.615\times 10^{26}$ Root discriminant $$38.41$$ Ramified primes $2,173$ Class number $15$ (GRH) Class group [15] (GRH) Galois group $\PSL(2,16)$ (as 17T6)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74)

gp: K = bnfinit(y^17 - y^16 - 4*y^15 + 2*y^14 + 54*y^13 - 6*y^12 - 36*y^11 + 16*y^10 + 714*y^9 + 1238*y^8 + 484*y^7 - 764*y^6 - 1084*y^5 + 520*y^4 + 668*y^3 - 776*y^2 + 382*y - 74, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74)

$$x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + \cdots - 74$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

## Invariants

 Degree: $17$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K);  oscar: degree(K) Signature: $[1, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$861526607800060221948166144$$ 861526607800060221948166144 $$\medspace = 2^{30}\cdot 173^{8}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$38.41$$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(OK))^(1/Degree(K));  oscar: (1.0 * dK)^(1/degree(K)) Ramified primes: $$2$$, $$173$$ 2, 173 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q$$ $\card{ \Aut(K/\Q) }$: $1$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{51\!\cdots\!19}a^{16}-\frac{29\!\cdots\!61}{51\!\cdots\!19}a^{15}-\frac{14\!\cdots\!39}{51\!\cdots\!19}a^{14}+\frac{87\!\cdots\!85}{51\!\cdots\!19}a^{13}+\frac{90\!\cdots\!97}{51\!\cdots\!19}a^{12}+\frac{21\!\cdots\!08}{51\!\cdots\!19}a^{11}+\frac{72\!\cdots\!54}{51\!\cdots\!19}a^{10}+\frac{31\!\cdots\!96}{51\!\cdots\!19}a^{9}+\frac{25\!\cdots\!45}{51\!\cdots\!19}a^{8}-\frac{49\!\cdots\!91}{51\!\cdots\!19}a^{7}-\frac{27\!\cdots\!52}{51\!\cdots\!19}a^{6}+\frac{22\!\cdots\!06}{51\!\cdots\!19}a^{5}-\frac{63\!\cdots\!14}{51\!\cdots\!19}a^{4}+\frac{14\!\cdots\!17}{51\!\cdots\!19}a^{3}+\frac{30\!\cdots\!20}{51\!\cdots\!19}a^{2}+\frac{22\!\cdots\!38}{51\!\cdots\!19}a+\frac{10\!\cdots\!37}{51\!\cdots\!19}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: Not computed Index: $1$ Inessential primes: None

## Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

## Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

 Rank: $8$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  oscar: torsion_units_generator(OK) Fundamental units: $\frac{44\!\cdots\!32}{51\!\cdots\!19}a^{16}-\frac{27\!\cdots\!92}{51\!\cdots\!19}a^{15}-\frac{19\!\cdots\!79}{51\!\cdots\!19}a^{14}+\frac{84\!\cdots\!50}{51\!\cdots\!19}a^{13}+\frac{24\!\cdots\!30}{51\!\cdots\!19}a^{12}+\frac{67\!\cdots\!12}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!79}{51\!\cdots\!19}a^{10}-\frac{36\!\cdots\!62}{51\!\cdots\!19}a^{9}+\frac{32\!\cdots\!07}{51\!\cdots\!19}a^{8}+\frac{67\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{45\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{26\!\cdots\!08}{51\!\cdots\!19}a^{5}-\frac{70\!\cdots\!44}{51\!\cdots\!19}a^{4}-\frac{96\!\cdots\!88}{51\!\cdots\!19}a^{3}+\frac{28\!\cdots\!28}{51\!\cdots\!19}a^{2}-\frac{17\!\cdots\!58}{51\!\cdots\!19}a+\frac{75\!\cdots\!83}{51\!\cdots\!19}$, $\frac{21\!\cdots\!47}{51\!\cdots\!19}a^{16}-\frac{12\!\cdots\!41}{51\!\cdots\!19}a^{15}-\frac{93\!\cdots\!11}{51\!\cdots\!19}a^{14}+\frac{43\!\cdots\!83}{51\!\cdots\!19}a^{13}+\frac{11\!\cdots\!10}{51\!\cdots\!19}a^{12}+\frac{37\!\cdots\!85}{51\!\cdots\!19}a^{11}-\frac{69\!\cdots\!45}{51\!\cdots\!19}a^{10}+\frac{39\!\cdots\!07}{51\!\cdots\!19}a^{9}+\frac{15\!\cdots\!48}{51\!\cdots\!19}a^{8}+\frac{33\!\cdots\!96}{51\!\cdots\!19}a^{7}+\frac{24\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{82\!\cdots\!39}{51\!\cdots\!19}a^{5}-\frac{29\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{20\!\cdots\!35}{51\!\cdots\!19}a^{3}+\frac{14\!\cdots\!31}{51\!\cdots\!19}a^{2}-\frac{11\!\cdots\!92}{51\!\cdots\!19}a+\frac{34\!\cdots\!07}{51\!\cdots\!19}$, $\frac{68\!\cdots\!28}{51\!\cdots\!19}a^{16}+\frac{16\!\cdots\!86}{51\!\cdots\!19}a^{15}-\frac{28\!\cdots\!39}{51\!\cdots\!19}a^{14}-\frac{21\!\cdots\!35}{51\!\cdots\!19}a^{13}+\frac{35\!\cdots\!80}{51\!\cdots\!19}a^{12}+\frac{40\!\cdots\!41}{51\!\cdots\!19}a^{11}+\frac{11\!\cdots\!07}{51\!\cdots\!19}a^{10}+\frac{96\!\cdots\!91}{51\!\cdots\!19}a^{9}+\frac{49\!\cdots\!40}{51\!\cdots\!19}a^{8}+\frac{14\!\cdots\!69}{51\!\cdots\!19}a^{7}+\frac{19\!\cdots\!93}{51\!\cdots\!19}a^{6}+\frac{12\!\cdots\!64}{51\!\cdots\!19}a^{5}+\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{4}-\frac{27\!\cdots\!16}{51\!\cdots\!19}a^{3}+\frac{36\!\cdots\!57}{51\!\cdots\!19}a^{2}-\frac{12\!\cdots\!34}{51\!\cdots\!19}a+\frac{23\!\cdots\!51}{51\!\cdots\!19}$, $\frac{32\!\cdots\!72}{51\!\cdots\!19}a^{16}-\frac{23\!\cdots\!33}{51\!\cdots\!19}a^{15}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{14}+\frac{89\!\cdots\!57}{51\!\cdots\!19}a^{13}+\frac{25\!\cdots\!11}{51\!\cdots\!19}a^{12}-\frac{10\!\cdots\!39}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{10}-\frac{27\!\cdots\!98}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!53}{51\!\cdots\!19}a^{8}-\frac{11\!\cdots\!88}{51\!\cdots\!19}a^{7}-\frac{42\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!55}{51\!\cdots\!19}a^{5}-\frac{45\!\cdots\!78}{51\!\cdots\!19}a^{4}-\frac{95\!\cdots\!35}{51\!\cdots\!19}a^{3}-\frac{65\!\cdots\!16}{51\!\cdots\!19}a^{2}-\frac{73\!\cdots\!26}{51\!\cdots\!19}a+\frac{16\!\cdots\!27}{51\!\cdots\!19}$, $\frac{15\!\cdots\!79}{51\!\cdots\!19}a^{16}-\frac{83\!\cdots\!62}{51\!\cdots\!19}a^{15}-\frac{67\!\cdots\!22}{51\!\cdots\!19}a^{14}+\frac{19\!\cdots\!51}{51\!\cdots\!19}a^{13}+\frac{84\!\cdots\!23}{51\!\cdots\!19}a^{12}+\frac{29\!\cdots\!81}{51\!\cdots\!19}a^{11}-\frac{50\!\cdots\!52}{51\!\cdots\!19}a^{10}+\frac{46\!\cdots\!34}{51\!\cdots\!19}a^{9}+\frac{11\!\cdots\!52}{51\!\cdots\!19}a^{8}+\frac{24\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!71}{51\!\cdots\!19}a^{6}-\frac{53\!\cdots\!23}{51\!\cdots\!19}a^{5}-\frac{19\!\cdots\!94}{51\!\cdots\!19}a^{4}+\frac{34\!\cdots\!29}{51\!\cdots\!19}a^{3}+\frac{12\!\cdots\!49}{51\!\cdots\!19}a^{2}-\frac{74\!\cdots\!71}{51\!\cdots\!19}a+\frac{20\!\cdots\!81}{51\!\cdots\!19}$, $\frac{12\!\cdots\!34}{51\!\cdots\!19}a^{16}-\frac{55\!\cdots\!88}{51\!\cdots\!19}a^{15}-\frac{55\!\cdots\!10}{51\!\cdots\!19}a^{14}-\frac{49\!\cdots\!28}{51\!\cdots\!19}a^{13}+\frac{69\!\cdots\!96}{51\!\cdots\!19}a^{12}+\frac{31\!\cdots\!77}{51\!\cdots\!19}a^{11}-\frac{31\!\cdots\!48}{51\!\cdots\!19}a^{10}+\frac{35\!\cdots\!03}{51\!\cdots\!19}a^{9}+\frac{91\!\cdots\!12}{51\!\cdots\!19}a^{8}+\frac{21\!\cdots\!91}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!64}{51\!\cdots\!19}a^{6}-\frac{46\!\cdots\!04}{51\!\cdots\!19}a^{5}-\frac{14\!\cdots\!99}{51\!\cdots\!19}a^{4}-\frac{14\!\cdots\!03}{51\!\cdots\!19}a^{3}+\frac{80\!\cdots\!99}{51\!\cdots\!19}a^{2}-\frac{59\!\cdots\!78}{51\!\cdots\!19}a+\frac{17\!\cdots\!29}{51\!\cdots\!19}$, $\frac{33\!\cdots\!74}{51\!\cdots\!19}a^{16}-\frac{88\!\cdots\!23}{51\!\cdots\!19}a^{15}-\frac{15\!\cdots\!99}{51\!\cdots\!19}a^{14}-\frac{19\!\cdots\!30}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!25}{51\!\cdots\!19}a^{12}+\frac{10\!\cdots\!04}{51\!\cdots\!19}a^{11}-\frac{12\!\cdots\!24}{51\!\cdots\!19}a^{10}+\frac{25\!\cdots\!97}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!80}{51\!\cdots\!19}a^{8}+\frac{59\!\cdots\!89}{51\!\cdots\!19}a^{7}+\frac{48\!\cdots\!06}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!87}{51\!\cdots\!19}a^{5}-\frac{48\!\cdots\!83}{51\!\cdots\!19}a^{4}-\frac{12\!\cdots\!77}{51\!\cdots\!19}a^{3}+\frac{37\!\cdots\!60}{51\!\cdots\!19}a^{2}-\frac{19\!\cdots\!86}{51\!\cdots\!19}a+\frac{73\!\cdots\!79}{51\!\cdots\!19}$, $\frac{36\!\cdots\!06}{51\!\cdots\!19}a^{16}-\frac{78\!\cdots\!91}{51\!\cdots\!19}a^{15}-\frac{62\!\cdots\!61}{51\!\cdots\!19}a^{14}+\frac{78\!\cdots\!32}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!98}{51\!\cdots\!19}a^{12}-\frac{13\!\cdots\!02}{51\!\cdots\!19}a^{11}+\frac{37\!\cdots\!44}{51\!\cdots\!19}a^{10}-\frac{86\!\cdots\!77}{51\!\cdots\!19}a^{9}+\frac{26\!\cdots\!17}{51\!\cdots\!19}a^{8}+\frac{31\!\cdots\!15}{51\!\cdots\!19}a^{7}-\frac{30\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{11\!\cdots\!90}{51\!\cdots\!19}a^{5}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{58\!\cdots\!75}{51\!\cdots\!19}a^{3}+\frac{39\!\cdots\!91}{51\!\cdots\!19}a^{2}-\frac{30\!\cdots\!87}{51\!\cdots\!19}a+\frac{22\!\cdots\!57}{51\!\cdots\!19}$ 4464450691457888219127032/518449026921810234250726519*a^16 - 2778077651056340703627492/518449026921810234250726519*a^15 - 19099343160650306364113179/518449026921810234250726519*a^14 + 840000348123006134290950/518449026921810234250726519*a^13 + 243626097347702830887753430/518449026921810234250726519*a^12 + 67759552090794232799747212/518449026921810234250726519*a^11 - 147594529874799854508116079/518449026921810234250726519*a^10 - 36133204068351849659180062/518449026921810234250726519*a^9 + 3202203210414220049184157507/518449026921810234250726519*a^8 + 6713751819624370883477867011/518449026921810234250726519*a^7 + 4572565643288289463236353338/518449026921810234250726519*a^6 - 2601143819586072033657002008/518449026921810234250726519*a^5 - 7031843738159010148818909944/518449026921810234250726519*a^4 - 962539560041004528037778688/518449026921810234250726519*a^3 + 2818164637943871543542195128/518449026921810234250726519*a^2 - 1701556418368547617748555358/518449026921810234250726519*a + 753870258585790541092256583/518449026921810234250726519, 21807114262001114193418947/518449026921810234250726519*a^16 - 12478962613728058427905741/518449026921810234250726519*a^15 - 93871049867252055582147211/518449026921810234250726519*a^14 + 4378359612658816967470483/518449026921810234250726519*a^13 + 1183818512001398316884742110/518449026921810234250726519*a^12 + 375488680681567271487083285/518449026921810234250726519*a^11 - 691419555286801464088251645/518449026921810234250726519*a^10 + 39793502062838552064680907/518449026921810234250726519*a^9 + 15573498264929675222952215248/518449026921810234250726519*a^8 + 33641701402804420490291569596/518449026921810234250726519*a^7 + 24003479781306459178168174355/518449026921810234250726519*a^6 - 8282768568802960438555996039/518449026921810234250726519*a^5 - 29128376247862596322639233247/518449026921810234250726519*a^4 - 2093340239503375426799419635/518449026921810234250726519*a^3 + 14043454754952792951881820431/518449026921810234250726519*a^2 - 11668980007201171541164764692/518449026921810234250726519*a + 3493001323678648651803439307/518449026921810234250726519, 6807316263347736197711928/518449026921810234250726519*a^16 + 1644718135120494275109286/518449026921810234250726519*a^15 - 28022360360928334904201139/518449026921810234250726519*a^14 - 21610416755385761438662235/518449026921810234250726519*a^13 + 353196750820526296316835980/518449026921810234250726519*a^12 + 401862204840515833247385841/518449026921810234250726519*a^11 + 110727974486214994430068007/518449026921810234250726519*a^10 + 96864175059792416980901791/518449026921810234250726519*a^9 + 4968771816827912292952664340/518449026921810234250726519*a^8 + 14417964725199345546031521469/518449026921810234250726519*a^7 + 19353976898566384896726636093/518449026921810234250726519*a^6 + 12828063805413987612388650464/518449026921810234250726519*a^5 + 1436926428934288367847636120/518449026921810234250726519*a^4 - 275376696257824771268076216/518449026921810234250726519*a^3 + 3698081061343389499541397257/518449026921810234250726519*a^2 - 1235252271638767828511873434/518449026921810234250726519*a + 231286364568997557179504551/518449026921810234250726519, 325772978888328922753772/518449026921810234250726519*a^16 - 2332028059572478195157833/518449026921810234250726519*a^15 - 2076517759100208201644247/518449026921810234250726519*a^14 + 8960758668820467182745357/518449026921810234250726519*a^13 + 25879770227137916038356811/518449026921810234250726519*a^12 - 108143820900599749795679839/518449026921810234250726519*a^11 - 146027666973307301970534320/518449026921810234250726519*a^10 - 27158525472672144104636298/518449026921810234250726519*a^9 + 231090400024750115865464653/518449026921810234250726519*a^8 - 1173326604813166143095850288/518449026921810234250726519*a^7 - 4247275270316821458870022538/518449026921810234250726519*a^6 - 6266294556557209603037776755/518449026921810234250726519*a^5 - 4595940222004033160610255178/518449026921810234250726519*a^4 - 952082239258945208945595835/518449026921810234250726519*a^3 - 65446657689732667907716016/518449026921810234250726519*a^2 - 732925405642264225906652726/518449026921810234250726519*a + 164209407965228746273363327/518449026921810234250726519, 15442180745624312526534179/518449026921810234250726519*a^16 - 8302649535143152334241462/518449026921810234250726519*a^15 - 67258135352961026094748622/518449026921810234250726519*a^14 + 1991699725933660549700851/518449026921810234250726519*a^13 + 840121564762739215573554423/518449026921810234250726519*a^12 + 292572981690985096541108881/518449026921810234250726519*a^11 - 508993745325468478256603152/518449026921810234250726519*a^10 + 46168377789770301005546534/518449026921810234250726519*a^9 + 11072745140146991838489672952/518449026921810234250726519*a^8 + 24269720052298977113461344211/518449026921810234250726519*a^7 + 17456832247686245979476065671/518449026921810234250726519*a^6 - 5357384579887323470115410023/518449026921810234250726519*a^5 - 19752787877096831280172763994/518449026921810234250726519*a^4 + 340471438365957011547248229/518449026921810234250726519*a^3 + 12148836636963571290185933449/518449026921810234250726519*a^2 - 7482621978539213829960316671/518449026921810234250726519*a + 2046079679417802978730096681/518449026921810234250726519, 12816263711183996506298934/518449026921810234250726519*a^16 - 5500712006779831271717188/518449026921810234250726519*a^15 - 55128804009581710942417810/518449026921810234250726519*a^14 - 4983307045611374061073228/518449026921810234250726519*a^13 + 691825566766543100779574696/518449026921810234250726519*a^12 + 315991277606627538908773377/518449026921810234250726519*a^11 - 318807140157446402379278548/518449026921810234250726519*a^10 + 35155130510720349834856803/518449026921810234250726519*a^9 + 9184756822494719070763372012/518449026921810234250726519*a^8 + 21079952562020856461420890791/518449026921810234250726519*a^7 + 17732077566812578278385186664/518449026921810234250726519*a^6 - 465596996761661216876943104/518449026921810234250726519*a^5 - 14536631465834807606342921399/518449026921810234250726519*a^4 - 1438774751754740286691115203/518449026921810234250726519*a^3 + 8088294769967018109853381199/518449026921810234250726519*a^2 - 5903975854379506921537345578/518449026921810234250726519*a + 1765516403022442175378704029/518449026921810234250726519, 3354074379222128372277674/518449026921810234250726519*a^16 - 889858522483991251557523/518449026921810234250726519*a^15 - 15815512437409584669141999/518449026921810234250726519*a^14 - 1908437195839031835926930/518449026921810234250726519*a^13 + 183119562528060292472499225/518449026921810234250726519*a^12 + 109727503285562280989946804/518449026921810234250726519*a^11 - 122366979299948848019838924/518449026921810234250726519*a^10 + 25027928726451208800162997/518449026921810234250726519*a^9 + 2346575274392929302995968880/518449026921810234250726519*a^8 + 5920752582048878595877574389/518449026921810234250726519*a^7 + 4896773393207362191807341206/518449026921810234250726519*a^6 - 624380622029945679773348087/518449026921810234250726519*a^5 - 4836004685627735175399066483/518449026921810234250726519*a^4 - 1247640344802977173168098777/518449026921810234250726519*a^3 + 3798198255632122682306544560/518449026921810234250726519*a^2 - 1999437758969872094709529686/518449026921810234250726519*a + 734264881612328059726664879/518449026921810234250726519, 368877631866614265508606/518449026921810234250726519*a^16 - 783403095225277713119991/518449026921810234250726519*a^15 - 629538891732819858439261/518449026921810234250726519*a^14 + 78054757550634795674832/518449026921810234250726519*a^13 + 18869413147164132311323498/518449026921810234250726519*a^12 - 13795080337073087994226802/518449026921810234250726519*a^11 + 3744194309540716388985944/518449026921810234250726519*a^10 - 86819335162187412427143077/518449026921810234250726519*a^9 + 260035635654685895319186117/518449026921810234250726519*a^8 + 311673458180276975210090015/518449026921810234250726519*a^7 - 306990998524667548611984655/518449026921810234250726519*a^6 - 1195270954512322766268836690/518449026921810234250726519*a^5 - 2017172352018197375497096447/518449026921810234250726519*a^4 - 581090801736428039525125375/518449026921810234250726519*a^3 + 395484407223711547007707191/518449026921810234250726519*a^2 - 305212630429169614825795687/518449026921810234250726519*a + 222309746828792703832780857/518449026921810234250726519 (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K|fUK(g): g in Generators(UK)];  oscar: [K(fUK(a)) for a in gens(UK)] Regulator: $$781146.759847$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 781146.759847 \cdot 15}{2\cdot\sqrt{861526607800060221948166144}}\cr\approx \mathstrut & 0.969680124809 \end{aligned} (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

## Galois group

$\SL(2,16)$ (as 17T6):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

 A non-solvable group of order 4080 The 17 conjugacy class representatives for $\PSL(2,16)$ Character table for $\PSL(2,16)$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R $17$ $17$ $17$ $17$ $15{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $17$ $17$ $15{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $17$ $17$ ${\href{/padicField/53.3.0.1}{3} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))]; # to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_K$for$p=7$in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac] ## Local algebras for ramified primes$p$LabelPolynomial$efc$Galois group Slope content $$2$$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$Deg$16$$16$$1$$30 $$173$$ \Q_{173}$$x + 171$$1$$1$$0Trivial[\ ] 173.4.2.1x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$173.4.2.1$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 173.4.2.1x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$173.4.2.1$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$$2$$2$$2$$C_2^2[\ ]_{2}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$15.861...144.240.a.a$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.b$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.c$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.d$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.e$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.f$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.g$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$15.861...144.240.a.h$15 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$1-1$* 16.861...144.17t6.a.a$16 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$10$17.861...144.51.a.a$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.68.a.a$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.68.a.b$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.120.a.a$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.120.a.b$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.120.a.c$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11$17.861...144.120.a.d$17 2^{30} \cdot 173^{8}$17.1.861526607800060221948166144.1$\PSL(2,16)$(as 17T6)$11\$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.